Users' Mathboxes Mathbox for Mario Carneiro < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  elmpps Structured version   Visualization version   GIF version

Theorem elmpps 33248
Description: Definition of a provable pre-statement, essentially just a reorganization of the arguments of df-mcls . (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
mppsval.p 𝑃 = (mPreSt‘𝑇)
mppsval.j 𝐽 = (mPPSt‘𝑇)
mppsval.c 𝐶 = (mCls‘𝑇)
Assertion
Ref Expression
elmpps (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝐽 ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐴 ∈ (𝐷𝐶𝐻)))

Proof of Theorem elmpps
Dummy variables 𝑎 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ot 4550 . . 3 𝐷, 𝐻, 𝐴⟩ = ⟨⟨𝐷, 𝐻⟩, 𝐴
2 mppsval.p . . . 4 𝑃 = (mPreSt‘𝑇)
3 mppsval.j . . . 4 𝐽 = (mPPSt‘𝑇)
4 mppsval.c . . . 4 𝐶 = (mCls‘𝑇)
52, 3, 4mppsval 33247 . . 3 𝐽 = {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))}
61, 5eleq12i 2830 . 2 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝐽 ↔ ⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))})
7 oprabss 7317 . . . 4 {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))} ⊆ ((V × V) × V)
87sseli 3896 . . 3 (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))} → ⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ ((V × V) × V))
92mpstssv 33214 . . . . . 6 𝑃 ⊆ ((V × V) × V)
109sseli 3896 . . . . 5 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → ⟨𝐷, 𝐻, 𝐴⟩ ∈ ((V × V) × V))
111, 10eqeltrrid 2843 . . . 4 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → ⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ ((V × V) × V))
1211adantr 484 . . 3 ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐴 ∈ (𝐷𝐶𝐻)) → ⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ ((V × V) × V))
13 opelxp 5587 . . . 4 (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ ((V × V) × V) ↔ (⟨𝐷, 𝐻⟩ ∈ (V × V) ∧ 𝐴 ∈ V))
14 opelxp 5587 . . . . 5 (⟨𝐷, 𝐻⟩ ∈ (V × V) ↔ (𝐷 ∈ V ∧ 𝐻 ∈ V))
15 simp1 1138 . . . . . . . . . 10 ((𝑑 = 𝐷 = 𝐻𝑎 = 𝐴) → 𝑑 = 𝐷)
16 simp2 1139 . . . . . . . . . 10 ((𝑑 = 𝐷 = 𝐻𝑎 = 𝐴) → = 𝐻)
17 simp3 1140 . . . . . . . . . 10 ((𝑑 = 𝐷 = 𝐻𝑎 = 𝐴) → 𝑎 = 𝐴)
1815, 16, 17oteq123d 4799 . . . . . . . . 9 ((𝑑 = 𝐷 = 𝐻𝑎 = 𝐴) → ⟨𝑑, , 𝑎⟩ = ⟨𝐷, 𝐻, 𝐴⟩)
1918eleq1d 2822 . . . . . . . 8 ((𝑑 = 𝐷 = 𝐻𝑎 = 𝐴) → (⟨𝑑, , 𝑎⟩ ∈ 𝑃 ↔ ⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃))
2015, 16oveq12d 7231 . . . . . . . . 9 ((𝑑 = 𝐷 = 𝐻𝑎 = 𝐴) → (𝑑𝐶) = (𝐷𝐶𝐻))
2117, 20eleq12d 2832 . . . . . . . 8 ((𝑑 = 𝐷 = 𝐻𝑎 = 𝐴) → (𝑎 ∈ (𝑑𝐶) ↔ 𝐴 ∈ (𝐷𝐶𝐻)))
2219, 21anbi12d 634 . . . . . . 7 ((𝑑 = 𝐷 = 𝐻𝑎 = 𝐴) → ((⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶)) ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐴 ∈ (𝐷𝐶𝐻))))
2322eloprabga 7318 . . . . . 6 ((𝐷 ∈ V ∧ 𝐻 ∈ V ∧ 𝐴 ∈ V) → (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))} ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐴 ∈ (𝐷𝐶𝐻))))
24233expa 1120 . . . . 5 (((𝐷 ∈ V ∧ 𝐻 ∈ V) ∧ 𝐴 ∈ V) → (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))} ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐴 ∈ (𝐷𝐶𝐻))))
2514, 24sylanb 584 . . . 4 ((⟨𝐷, 𝐻⟩ ∈ (V × V) ∧ 𝐴 ∈ V) → (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))} ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐴 ∈ (𝐷𝐶𝐻))))
2613, 25sylbi 220 . . 3 (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ ((V × V) × V) → (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))} ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐴 ∈ (𝐷𝐶𝐻))))
278, 12, 26pm5.21nii 383 . 2 (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))} ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐴 ∈ (𝐷𝐶𝐻)))
286, 27bitri 278 1 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝐽 ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐴 ∈ (𝐷𝐶𝐻)))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  w3a 1089   = wceq 1543  wcel 2110  Vcvv 3408  cop 4547  cotp 4549   × cxp 5549  cfv 6380  (class class class)co 7213  {coprab 7214  mPreStcmpst 33148  mClscmcls 33152  mPPStcmpps 33153
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-ot 4550  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-iota 6338  df-fun 6382  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpst 33168  df-mpps 33173
This theorem is referenced by:  mthmpps  33257  mclspps  33259
  Copyright terms: Public domain W3C validator